Clusters of repetition roots

去年，我们成功地扩展了锚位技术，以表示指数大于2的重复。利用这一点，我们证明了我们的聚类大小猜想适用于具有任意整数指数的重复根的单链。我们还证明了单链的界是最优的，并给出了在界内允许的任何簇大小组合的簇序列的建设性证明。这些结果已发表在DCFS 2022中。去年的一个重大进展是李和布莱克利用Rauzy图的性质解决了Fraenkel-Simpson猜想。我们试图将他们的论点扩展到所谓的扩展Rauzy图，以证明我们在一般情况下的聚类猜想。介绍这些结果的论文目前正在编写中。

In the last year we managed to extend the technique of anchor positions to represent repetitions with exponents higher than 2. Using this, we showed that our cluster size conjecture holds for single chains of repetition roots with arbitrary integer exponent.We also showed that the bounds given for single chains are optimal by a constructive proof yielding sequences of clusters for any combination of cluster sizes allowed within the bounds. These results have been published in DCFS 2022.A big development from last year was the solution of the Fraenkel-Simpson conjecture by Li and Brlek using properties of Rauzy graphs. We tried to extend their arguments to so called extended Rauzy graphs to prove our clusters conjecture in the general case. The paper presenting those results is under work as of now.